(a) An integer N is to be selected at random from {1, 2, . . . , (10)3} in the sense that each integer has the same probability of being selected. What is the probability that N will be divisible by 3? by 5? by 7? by 15? by 105? How would your answer change if (10)3 is replaced by (10)k as k became larger and larger?
(b) An important function in number theory€”one whose properties can be shown to be related to what is probably the most important unsolved problem of mathematics, the Riemann hypothesis€”is the MÓ§bius function μ(n), defined for all positive integral values n as follows: Factor n into its prime factors. If there is a repeated prime factor, as in 12 = 2 · 2 · 3 or 49 = 7 · 7, then μ(n) is defined to equal 0. Now let N be chosen at random from {1, 2, . . . (10)k}, where k is large. Determine P{μ(N) = 0} as k†’ˆž.
To compute P{μ(N) ‰  0}, use the identity

where Pi is the ith-smallest prime. (The number 1 is not a prime.)

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